If you think about, while the motor is running there is no difference between what is inside the motor. As soon as it starts any air is immediately burnt or expelled, so what is left is the burning propellant and gases from the burn. The rate that the gass leaves the engine is pretty really dependant of the throat size of the nozzle and the internal pressure, external pressure is insignificant. The startup is what is different, it will take fractionally longer to start and pressurise. Technically the shut down will be a little different too, but only once the thrust drops down to a fraction of a percent so we really don't care much about that.

Yes the thrust produced is lower because there is no air to push against, but then again the front of the rocket does not have any air to stop it moving forward and there is not drag down the side of the rocket etc. So the overall efficiency does go up.

As I understand it the main difference in engine design for a vacuum is the expansion rate of the nozzle, once the gass exits the engine it expands much more quickly. So engine designers modify the bell shape of the nozzle to optimise it for vacum.

Yes the thrust produced is lower because there is no air to push against

Rocket engines work because of Newton's third law, not because they push on anything. Throw something out the back and an equal reactive force works forwards.

The air pressure does constrain the expansion of the exhaust to a narrower angle, but does not contribute to the thrust. As you alluded to in your second paragraph there is air pressure all around the rocket, so air pressure on the front of the rocket cancels any pressure assistance from the rear.

As I understand it the main difference in engine design for a vacuum is the expansion rate of the nozzle, once the gass exits the engine it expands much more quickly. So engine designers modify the bell shape of the nozzle to optimise it for vacum.

Correct. The faster the particles exit the engine nozzle, the more they have been accelerated by the combustion and the greater the reaction force that propels the rocket. The flow, sonic at the throat, is accelerated supersonic by expansion. The bigger the expansion ratio of the nozzle the more thrust can be extracted from the burn, because the exhaust mass is faster (when comparing same mass-flow engines).

Engines designed for lower in the atmosphere have a lower expansion rate as the gas stream is prevented from expanding by the air pressure surrounding it. That means that the expansion bells can be smaller as well.

Vacuum engines can have seriously large expansion ratios (up to about 300 IIRC is practical) but the thing that limits them is the size and mass that they consume as that needs to be lifted by the previous stage(s). The bigger the ratio the better, until it starts eating in the the launch budget elsewhere.

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I would recommend this book to anyone who wants to know about all the intricacies of propulsion. There are good chapters on chemical and electric propulsion, as well as a very thorough look at nuclear thermal rocket engines. If you search the web for it there may even be a pdf of it somewhere

Found this discussion on the comparison be tweeted SpaceX normal merlin and vacuum rated engines. Some great side by side pictures of the engines and also a second picture of the massive nozzle extension for the vacuum rated engine.